Transcendental Kähler Cohomology Classes

  • Dan Popovici

    Université Paul Sabatier, Toulouse, France


Associated with a real, smooth, -closed -form of possibly non-rational de Rham cohomology class on a compact complex manifold is a sequence of asymptotically holomorphic complex line bundles on equipped with -connections for which . Their study was begun in the thesis of L. Laeng. We propose in this non-integrable context a substitute for Hörmander's familiar -estimates of the -equation of the integrable case that is based on analysing the spectra of the Laplace–Beltrami operators associated with . Global approximately holomorphic peak sections of are constructed as a counterpart to Tian's holomorphic peak sections of the integral-class case. Two applications are then obtained when is strictly positive: a Kodaira-type approximately holomorphic projective embedding theorem and a Tian-type almost-isometry theorem for compact Kähler, possibly non-projective, manifolds. Unlike in similar results in the literature for symplectic forms of integral classes, the peculiarity of lies in its transcendental class. This approach will be hopefully continued in future work by relaxing the positivity assumption on .

Cite this article

Dan Popovici, Transcendental Kähler Cohomology Classes. Publ. Res. Inst. Math. Sci. 49 (2013), no. 2, pp. 313–360

DOI 10.4171/PRIMS/107